ON LINEAR DYNAMICS OF THE BACKWARD SHIFT OPERATORS Текст научной статьи по специальности «Математика»

ON LINEAR DYNAMICS OF THE BACKWARD SHIFT

OPERATORS

1Xasanova Kumush Erkin qizi, 2Khakimov Otabek Norbuta ugli

University of Exact and Social Sciences, Uzbekistan, 2V.I.Romanovskiy Institute of

Mathematics, Uzbekistan https://doi.org/10.5281/zenodo.13894893

Abstract. Linear dynamics is a rapidly evolving branch of functional analysis, which was probably born in 1982 with the Toronto PhD thesis of C. Kitai [3]. It has become rather popular, thanks to the efforts of many mathematicians (see [1], [2]). In particular, hypercyclicity and supercyclicity of weighted bilateral shifts were characterized by Salas [4], [5]. In [6] it has been proved that there exists a bounded linear operator T satisfying Kitai Criterion on each separable infinite-dimensional Banach space. We are going to study dynamics of linear operators defined on topological vector space over non-Archimedean valued fields.

Keywords: T-orbit, hypercyclic, supercyclic, cyclic, linear dynamics, non-Archimedean Banach space.

Kalit so'zlar: T-orbita, gipersiklik, supersiklik, siklik, chiziqli dinamika, noarximed Banax

fazosi.

Ключевые слова: T-орбита, гиперциклический, суперциклический, циклический, линейная динамика, неархимедово банахово пространство.

Let X and Y be topological vector spaces over non-Archimedean valued field К By L(X, Y) we denote the set of all continuous linear operators from X to Y. If X = Y then L(X, Y) is denoted by L(X). In what follows, we use the following terminology: T is a linear continuous operator on X means that T E L(X). The T- orbit of a vector x E X, for some operator T E L(X), is the set

0(x,T):= (Tn(x);n E Z+).

An operator T E L(X) is called hypercyclic if there exists a vector x EX such that its Torbit is dense in X. The corresponding vector x is called T- hypercyclic, and the set of all T-hypercyclic vectors is denoted by HC(T). Similarly, T is called supercyclic if there exists a vector x EX such that whose projective orbit

K- 0(x,T): = (ATn(x);n E Ъ+, X E K)

is dense in X. The vector x EX with a dense projective Г-orbit, is called T-supercyclic vector and the set of all Г-supercyclic vectors is denoted by SC(T). Finally, we recall that T is called cyclic if there exists x EX such that

ЩГ]ж: = span 0(x,T) = (P(T)(x): P polynomial )

is dense in X and such kind of vectors are called Г-cyclic vectors. The set of all T-cyclic vectors is denoted by CC(T).

We stress that the notion of hypercyclicity makes sense only if the space X is separable. Note that one has

HC(T) с SC(T) с CC(T).

Moreover, if T is a hypercyclic operator on a Banach space then \\ T ||> 1.

Now we show that hypercyclicity turns out to be a purely infinite-dimensional

phenomenon.

Proposition 1. Let X Ф (0) be a finite-dimensional vector space over non-Archimedean field K. Then each operator T E L(X) is not hypercyclic.

Now we are going to sufficient and necessary conditions to hypercyclicity of linear operators on F-spaces. In what follows, by F-space we mean a topological vector space X which is metrizable and complete over a non-Archimedean field.

We start with the well-known equivalence between hypercyclicity and topological transitivity. We recall that an operator T is called topological transitive if for each pair of nonempty open sets U and V there exists n EN such that Tn(U) nV ^ 0.

Theorem 1. Let X be a separable F-space and T E L(X). Then T is hypercyclic iff it is topologically transitive.

We are going to study some basic properties of cyclic/supercyclic operators on c0 space which have not an analog for Archimedean case. In what follows, we always assume that c0 is a separable space. Note that the separability of c0 is equivalent to the separability of K. For every integer n we denote by en is a unit vector such that only the n-th coordinate equals to one and others are zero.

Lemma 1. Let T be a cyclic operator on c0. If x E c0 satisfies the following condition r(Tn(x))i = xi,

SMS w for some i ^ j, Vn E N,

(Tn(x))j=xj, J'

then CC(T).

Lemma 2. Let T be a supercyclic operator on c0. Assume that for a given x E c0, there exists an integer N > 0 such that

(l(Tn(x))il = l(Tm(x))il, . .

{I(Tn(x))jl = l(Tm(x))j\, f0r SOme l*J, Vn,™>N,

then SC(T).

Note that we have essentially used a non-Archimedean norm's property in the proofs of Lemma 1 and Lemma 2. For this reason, we can say that proved Lemmas are not hold in Archimedean case.

Now we are going to study dynamics of generalized weighted backward shift operator on non-Archimedean c0 space. Let us consider an infinite dimensional upper-triangular matrix W = (wi,j)Tj=i over a non-Archimedean field K, such that

supflWfjl} < wkil = 0, Vk> I. (1)

ij

For a given W with (1.4.1), we define the following linear operator on c0 by

(0, if n = 1;

Bw(en) = [Y1}— Wj,nej, if n>2. (2)

The linear operator (2) is called the generalized weighted backward shift operator. Recall that if matrix W has the following extra condition wk l = 0 for all k ^ I + 1, then the corresponding linear operator Bw is reduced to weighted backward shift. In this setting, the operator acts as follows: Bw(e1) = 0 and Bw(en) = wn_1en_1 if n > 2, where wn-1: = wn_ln is called a weighted backward shift. Here, w = (wn)nEN. The operator Bw is called a backward shift if wn-1 = 1 for all n > 1, and such a shift is denoted by B.

In many areas of mathematics, an operator I + T appears, where I is an identity and T is a given operator. In this section, we consider the generalized weighted backward shift operator Bw instead of T. Our aim is going to study the supercyclicity of such types of operators on c0. For a given W with (1), we denote

TW": = l + B-ffl,

that is, for any x E c0

(Tw(x))k = xk + Z?=k+1 wki]Xj, vk e N.

Theorem 2. Let W be a matrix given by (1). If sup(|w1y|) = 0, then the following

statements are equivalent:

(i) Tw is supercyclic on c0;

(ii) Tw, is hypercyclic on c0, where W = (w[j)™j=1 and w'j = wi+1j+1, Vi,j e N. REFERENCES

1. Godefroy G., Shapiro J.H., Operators with dense, invariant, cyclic vector manifolds. // J. Funct. Anal. 1991, 9 8, pp.229-269.

2. Grosse-Erdmann K.-G., Universal families and hypercyclic vectors. Bull. Am. Math. Soc. 1999, 36, pp.345-381.

3. Kitai C., Invariant Closed Sets for Linear Operators. // Ph.D Thesis, University of Toronto, Toronto, Canada, 1982.

4. Salas H., Hypercyclic weighted shifts. // Trans. Am. Math. Soc. 1995, 347, pp. 993-1004.

5. Salas H., Supercyclicity and weighted shifts. // Stud. Math. 1999, 135, pp.55-74.

6. Shkarin S., The Kitai Criterion and backward shifts. // Proc. Am. Math. Soc. 2006, 136, pp.1659-1670.